After this week, you will be able to:

1. Navigate and use the RStudio interface effectively
2. Execute basic R functions and understand their syntax
3. Feel confident explaining basic statistical concepts like samples and populations
4. Understand and explain measures of central tendency (mean, median, mode) clearly and without mathematical jargon
5. Master different measures of spread (range, IQR, variance, standard deviation) through practical examples
6. Calculate statistical measures using both R and Excel
7. Choose appropriate statistical measures for your biological data and explain your choices

Quick checklist

By now you should have…

• Installed R
• Installed RStudio
• Created one (or two) documents in Markdown using Quarto

From calculators to computers

Statistical software in the 1970s

R in today’s world

• Leading tool in data science and statistics (although Python leads in majority of machine learning workflows)
• Over 22,000 packages on CRAN – extensive statistical capabilities
• Integration with other modern tools: Python, HTML, Javascript, Excel, AJAX…
• Meets modern academic standards of reproducibility and increasingly preferred by statisticians

RStudio IDE. Source: Januar Harianto

Your RStudio workspace

Left: all input, Right: all output

We will always work in Quarto

• A Markdown-based authoring tool
• Allows you to write reproducible documents with code and text
• Outputs to various formats: HTML, PDF, Word, and more
• Quarto gallery

How Quarto works

• Write your content in Markdown (left panel)
• View the output in Preview (right panel)
• Quick demo (only available in the live lecture)

Essential Resources

• Read “A brief R guide for surviving ENVX1002” by Dr. Geoffrey Mazue
  • Available in the Tool Kit section on Canvas
  • Contains essential R programming tips and best practices for this unit
• Use the Help panel in RStudio

Population vs Sample

Populations and their samples

Disclaimer: use of GenAI for content generation in this slide.

How well does a sample represent the population?

It depends:

• Sample size: Larger samples are more likely to represent the population
• Sampling method: Random samples are more likely to be representative
• Population variability: More variability means larger samples are needed

In reality, we often have to balance these factors due to time, cost, and practical constraints.

Samples vary

Different samples give different results – suppose we have a population of 1000 trees and we randomly sample 6 tree heights. If this is done 3 times, it is likely that the samples will be different.

We have code to demonstrate this but just focus on the results for now:

set.seed(258) 
population <- rnorm(1000, mean = 12, sd = 5)

# create samples
sample1 <- round(sample(population, size = 6), 1)
sample2 <- round(sample(population, size = 6), 1)
sample3 <- round(sample(population, size = 6), 1)
# show samples
for (i in 1:3) {
   cat(sprintf("Sample %d: ", i), get(paste0("sample", i)), "\n")
}

Sample 1:  13.7 14.6 14.8 9.6 6.5 10 
Sample 2:  7.6 6.1 9.9 10.1 12.5 14.9 
Sample 3:  9.8 7.9 18.4 19.1 7 26.1 

Are the samples different? How different are they?

Descriptive statistics

We use descriptive statistics to summarise and describe data, helping us compare and contrast.

1. Measures of central tendency – describe the “typical” value in a sample
   • mean, median, mode
2. Measures of spread – describe how much the data varies
   • standard deviation, variance (commonly used)
   • range, quartiles, IQR (for unique cases)
3. Measures of uncertainty – describe how confident we are in our estimates
   • standard error
   • confidence intervals

Mean – also known as the average

• Add up all your numbers
• Divide by how many numbers you have

Mean in Excel

Mean in R

I’ll show you how to do this in RStudio.

Step 1: Create a vector of values

In R, we store data in vectors (lists of values):

# Create a vector of test scores
scores <- c(80, 85, 90, 95)

# Display the scores
scores

[1] 80 85 90 95

Step 2: Calculate

# Sum divided by count
sum(scores) / length(scores)

[1] 87.5

# The mean() function does the work for us
mean(scores)

[1] 87.5

Median – the middle value

The median is the middle number when your data is in order:

1. First, put your numbers in order
2. Find the middle value
3. If you have an even number of values, take the average of the two middle numbers

Median in Excel

Excel provides two main ways to find the median:

1. Using MEDIAN function

   =MEDIAN(A1:A9)

   • Type =MEDIAN(
   • Select your data range
   • Press Enter

2. Alternative method

   • Sort your data first (use the Sort functionality in the Data tab)
   • Find middle value(s)
   • If even number of values, average the middle two

Median in R

R does all the ordering and finding the middle for us:

# House prices
prices <- c(450, 1100, 480, 460, 470, 420, 1400, 450, 470)

# Find median
median(prices)

[1] 470

# Compare with mean
mean(prices)

[1] 633.3333

Mode – most frequent value

Mode in Excel

Excel provides several methods to find the mode but the simplest is to use the MODE function:

=MODE(A1:A10)

• Type =MODE(
• Select your data range
• Press Enter

Mode in R

There is no built-in function to calculate the mode, so we use the modeest package:

if(!require("modeest")) install.packages("modeest")
library(modeest)

df <- c(1, 2, 3, 3, 4, 5, 5, 5, 6)
mlv(df, method = "mfv")  # most frequent value

[1] 5

freq_table <- table(df) # Count frequencies of each value
# Find which value(s) appear most often
modes <- as.numeric(names(freq_table[freq_table == max(freq_table)]))
modes

[1] 5

sorted_df <- sort(df) # Sort the vector first
runs <- rle(sorted_df) # Use run-length encoding to find sequences
modes <- runs$values[runs$lengths == max(runs$lengths)] # Find the value(s) with max length
modes

[1] 5

unique_vals <- factor(df) # Create a factor of unique values
counts <- tapply(df, unique_vals, length) # Count occurrences using tapply
modes <- as.numeric(names(counts[counts == max(counts)])) # Find which values have the maximum count
modes

[1] 5

A biological example

Source: Adobe Stock # 85659279

Imagine sampling seagrass blade lengths from two different sites in a marine ecosystem, and they have the same mean length of 15.2 cm. Are both sites the same?

Comparing Different Measures

# Plot seagrass lengths
library(ggplot2)
library(patchwork)

seagrass_protected <- c(15.2, 15.0, 15.3, 15.1, 15.2)
seagrass_exposed <- c(12.0, 18.0, 14.5, 16.5, 15.0)

# Create plots for both sites
p1 <- ggplot() +
   geom_point(aes(x = 1:5, y = seagrass_protected), size = 3) +
   geom_hline(yintercept = mean(seagrass_protected), linetype = "dashed", color = "red") +
   labs(title = "Site A: Protected Bay", x = "Measurement", y = "Length (cm)") +
   ylim(10, 20)

p2 <- ggplot() +
   geom_point(aes(x = 1:5, y = seagrass_exposed), size = 3) +
   geom_hline(yintercept = mean(seagrass_exposed), linetype = "dashed", color = "red") +
   labs(title = "Site B: Wave-exposed Coast", x = "Measurement", y = "Length (cm)") +
   ylim(10, 20)

# Combine plots side by side
p1 + p2

Why do we need measures of spread?

• Central tendency (mean, median, mode) only tells part of the story
• Spread tells us how much variation exists in our data
• Different measures of spread tell us different things:
  • Range: Overall spread of data
  • IQR: Spread of middle 50% of data
  • Variance: Average squared deviation from mean
  • Standard deviation: Average deviation in original units

Range – The simplest measure of spread

# Create our seagrass data
seagrass_protected <- c(15.2, 15.0, 15.3, 15.1, 15.2)  # Protected bay
seagrass_exposed <- c(12.0, 18.0, 14.5, 16.5, 15.0)    # Wave-exposed coast

# Calculate ranges
cat("Protected bay range:", diff(range(seagrass_protected)), "cm\n")

Protected bay range: 0.3 cm

cat("Wave-exposed range:", diff(range(seagrass_exposed)), "cm\n")

Wave-exposed range: 6 cm

Interquartile range (IQR): The middle 50%

The IQR tells us how spread out the middle 50% of our data is:

# Get quartiles for protected bay
quantile(seagrass_protected)

  0%  25%  50%  75% 100% 
15.0 15.1 15.2 15.2 15.3 

• 25% of data below Q1 (1st quartile)
• 75% of data below Q3 (3rd quartile)
• IQR = Q3 - Q1

Why use IQR?

• Ignores extreme values
• Works with skewed data
• More stable than range

Comparing Sites Using IQR

# Compare IQRs
pbay <- IQR(seagrass_protected)
pbay

[1] 0.1

exbay <- IQR(seagrass_exposed)
exbay

[1] 2

• Protected bay IQR: 0.1 cm
• Wave-exposed IQR: 2 cm

Variance: a detailed measure of spread

Variance measures how far data points are spread from their mean by:

1. Finding how far each point is from the mean
2. Squaring these distances (to handle negative values)
3. Taking the average of these squared distances

Why use variance?

• Uses all data points (unlike IQR)
• Less sensitive to outliers than range
• Shows total spread in both directions

Calculating Variance in R

# Calculate variance for both sites
cat("Protected bay variance:", var(seagrass_protected), "cm²\n")

Protected bay variance: 0.013 cm²

cat("Wave-exposed variance:", var(seagrass_exposed), "cm²\n")

Wave-exposed variance: 5.075 cm²

Standard deviation: a more interpretable measure

Standard deviation (SD, or \sigma for population, s for sample) is the square root of variance:

• Tells us the “typical distance” from the mean
• Easy to understand - similar to saying “± value” after a mean
• Small SD means values cluster closely around mean
• Large SD means values are more spread out

When and why to use it

• Values are in the same units as your data (unlike variance)
• Perfect for describing natural variation (height, weight, temperature)
• Used in many statistical tests
• Great for comparing different groups or datasets

Interpreting standard deviation (with R)

We can describe our seagrass lengths using mean ± standard deviation:

# Protected bay
mean_p <- mean(seagrass_protected)
sd_p <- sd(seagrass_protected)
cat("Protected bay:", round(mean_p, 1), "±", round(sd_p, 2), "cm\n")

Protected bay: 15.2 ± 0.11 cm

# Wave-exposed
mean_e <- mean(seagrass_exposed)
sd_e <- sd(seagrass_exposed)
cat("Wave-exposed:", round(mean_e, 1), "±", round(sd_e, 2), "cm\n")

Wave-exposed: 15.2 ± 2.25 cm

Comparing spread measures

Key Observations

• Wave-exposed site shows consistently more variation
• Each measure gives a different perspective
• Choose based on your data and goals
• Standard deviation is most commonly used in research papers

Range and IQR in Excel

Common Excel functions for measuring spread:

1. Range: Use MAX() and MIN()

=MAX(A1:A10) - MIN(A1:A10)

2. Quartiles and IQR: Use QUARTILE.INC()

For Q1: =QUARTILE.INC(A1:A10, 1)
For Q3: =QUARTILE.INC(A1:A10, 3)
For IQR: =QUARTILE.INC(A1:A10, 3) - QUARTILE.INC(A1:A10, 1)

Variance and standard deviation in Excel

Statistical functions for variance and standard deviation:

1. Sample Variance: Use VAR.S()

=VAR.S(A1:A10)

2. Sample Standard Deviation: Use STDEV.S()

=STDEV.S(A1:A10)

Core Reading

• Quinn & Keough (2024). Experimental Design and Data Analysis for Biologists. Cambridge University Press. Chapter 2: Things to know before proceeding.
• Canvas site for lecture notes and additional resources